If we had mastered a drive able to safely accelerate ships to one quarter the speed of light, it wouldn't take that long to get to Alpha Centauri. Especially in ext. hibernation. BTW, when I say distance, I mean estimated... And, this is only an example made to point out the ease with which earth's population could leave and arrive somewhere easily... also, imagine if they headed for a star cluster... where the entire cluster is no more than 10 or 20 lightyears across... a "minigalaxy"... you could start with one system and there'd be many more within 1 or 2 lightyears. earth is in the boondocks. (Though the problem is is the conditions of a star cluster are hostile for human life in terms of radiation, and resource poor) So, don't bash me on the incorrect reporting of the distance to Alpha Centauri... it was only an example... and anyway, .15 LY is NOTHING.
Here are some very basic calculations: REMEMBER, I USED METERS. NOT MILES. 1 Meter is equal to ABOUT 3.1 feet. Also, when I say "AT", I mean "The time it would take to get there traveling at".
SOL: 299,792,458 m/s (meters per second) (AKA a Lightsecond, as heard on Andromeda)
Distance to Alpha Centauri: 4.2 [or 4.36, but I used 4.2] Lightyears (lightyear = 1 year of travel at SOL)
(distance traveled @)SOL: 9,454,254,955,488,000 m/y (meters per year)
1/4 SOL: 2,363,563,738,872,000 m/y
FOR COMPARISON, the speed of sound at Sea Level (yes, it's different at different altitudes)
MACH 1: 10,731,385,440 meters
MACH 10: 107,313,854,400 m/y
MACH 20: 214,627,708,800 m/y
MACH 30: 321,941,563,200 m/y
MACH 100: 1,073,138,544,000 m/y
37,000 years at merely Mach 100? Not bad... but nice to know it'll only take 16.8 @ a quarter the speed of light. I know what colony ship I'M riding on.
Now, I've no idea what effect Time Dilation (what happens when you travel at speeds approaching the speed of light) would have on this, but I believe it stands as a good theory as to the size of the Firefly Universe, and how we got there...
A relativistic correction for time is easily solved for. the change in experienced time is equal to the change in original time divided by the Lorentz factor. The lorentz factor is the corrective formula applied to all relativistic calculations, the square root of 1 minus the velocity squared divided by the speed of light squared. If you were to attempt to plug this into a spreadsheet, the formula would be ((1-(velocity^2/(9*10^16)))^0.5) In any case, for a travel time of 16.8 years to the Centauri system, the experienced time would be 16.26 years. Not much of a difference.
However, all of these scenarios ignore the acceleration time. It takes time to get up to speed, and unfortunately, the only engines we have concieved of that are efficient enough to allow intersteller travel also have extremely low accelerations. So the transit time will be far greater then what is shown here.
Back of the envelope calcs: to reach 1/4 c (quarter light speed) at 1 g (earth gravity, typical hard sci-fi acceleration), you'd only need about 88 days. calculation(approve sites) So, speed up to 1/4 c, slow down at end, you only add at most 176 days, or about 0.5 years. Reasonable, given fancy drive that can sustain acceleration (i.e. firefly drive), but don't need inertial dampening. By the way, the rotating jets would on firefly would work great to maintain 1 g acceleration and give gravity oriented properly for the passengers. Of course, the tail seems to be where the major engine is.